The focus of this article lies on the notion of the edge version of doubly resolving sets (EVDRSs) in circulant networks. EVDRSs refer to unique edge subsets that are necessary for identifying individual edges in a network and distinguishing them based on their edge distances to the elements of the EVDRS. The main objectives were to define the minimal size of EVDRSs for circulant networks $C_n(1,2)$ and to investigate their basic properties. The systematic research helped to achieve a new understanding of the existence, construction, and characterization of EVDRSs in circulant networks $C_n(1,2)$. It is established that the EVDRSs in the circulant network $C_n(1,2)$ are finite and are bounded by the order of the network. Among the numerous implications of these findings are those that refer to the design and optimization of distributed sensor networks, improving communication and network protocols, as well as tracking the spread of infectious diseases and epidemics over social networks. The application of the identified methodology helps improve the process of network optimization which contributes to the development of more effective and robust circulant-based structures.